Skip to main content
SpringerLink
Log in

Spectra of compact locally symmetric manifolds of negative curvature

  • Published:
Inventiones mathematicae Aims and scope

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

References

  1. Araki, S.: On root systems and an infinitesimal classification of irreducible symmetric spaces. J. of Math., Osaka City Univ.13, 1–34 (1962)

    Google Scholar 

  2. Bérard, P. H.: On the wave equation on a compact Riemannian manifold without conjugate points. Math. Z.155, 249–276 (1977)

    Google Scholar 

  3. Borel, A.: Compact Clifford-Klein forms of symmetric spaces. Topology2, 111–122 (1963)

    Google Scholar 

  4. Colin de Verdière, Y.: Spectre conjoint d'opérateurs pseudo-différentiels qui commutent, I. Le cas non intégrable. Preprint, Institut Fourier, St. Martin d'Hères. (1978)

  5. DeGeorge, D.L., Wallach, N.R.: Limit formulas for multiplicities inL 2 (Γ/G). Ann. of Math.107, 133–150 (1978)

    Google Scholar 

  6. Duistermaat, J.J., Guillemin, V.W.: The spectrum of positive elliptic operators and periodic bicharacteristics. Invent. Math.29, 39–79 (1975)

    Google Scholar 

  7. Gangolli, R.: Asymptotic behaviour of spectra of compact quotients of certain symmetric spaces. Acta Math.121, 151–192 (1968)

    Google Scholar 

  8. Gangolli, R.: On the Plancherel formula and the Paley-Wiener theorem for spherical functions on semisimple Lie groups. Ann. of Math.93, 150–165 (1971)

    Google Scholar 

  9. Gangolli, R.: Spectra of discrete uniform subgroups of semisimple Lie groups. In: Symmetric Spaces. Short Courses Presented at Washington University, pp. 93–117. New York: Marcel Dekker 1972

    Google Scholar 

  10. Gangolli, R., Warner, G.: On Selberg's trace formula. J. Math. Soc. Japan27, 327–343 (1975)

    Google Scholar 

  11. Gel'fand, I.M.: Automorphic functions and the theory of representations. Proc. Int. Congr. of Math., Stockholm, 1962, pp. 74–85. Stockholm: Almquist & Wiksell 1963

    Google Scholar 

  12. Gel'fand, I.M., Graev, M.I.: Geometry of homogenoues spaces, representations of groups in homogeneous spaces, and related questions of integral geometry, I. Trudy. Moskov. mat. Obšč.8, 321–390 (1959)

    Google Scholar 

  13. Gindikin, S.G., Karpelevič, F.I.: Plancherel measure of Riemannian symmetric spaces of nonpositive curvature. Dokl. Akad. Nauk SSSR145, 252–255 (1962)

    Google Scholar 

  14. Harish-Chandra: On the Plancherel formula for the right-invariant functions on a semisimple Lie group. Proc. Nat. Acad. Sci. U.S.A.40, 200–204 (1954)

    Google Scholar 

  15. Harish-Chandra: The Plancherel formula for complex semisimple Lie groups. Trans. Amer. Math. Soc.76, 485–528 (1954)

    Google Scholar 

  16. Harish-Chandra: Differential operators on a semisimple Lie algebra. Amer. J. Math.79, 87–120 (1957)

    Google Scholar 

  17. Harish-Chandra: A formula for semisimple Lie groups. Amer. J. Math.79, 733–760 (1957)

    Google Scholar 

  18. Harish-Chandra: Spherical functions on a semisimple Lie group. I Amer. J. Math.80, 241–310 (1958)

    Google Scholar 

  19. Harish-Chandra: Spherical functions on a semisimple Lie group. II. Amer. J. Math.80, 553–613 (1958)

    Google Scholar 

  20. Harish-Chandra: Invariant eigendistributions on a semisimple Lie group. Trans. Amer. Math. Soc.119, 457–508 (1965)

    Google Scholar 

  21. Harish-Chandra: Discrete series for semisimple Lie groups, II. Explicit determination of the characters. Acta Math.116, 1–111 (1966)

    Google Scholar 

  22. Harish-Chandra: Harmonic analysis on real reductive groups, I. The theory of the constant term. J. of Functional Analysis19, 104–204 (1975)

    Google Scholar 

  23. Hejhal, D.A.: The Selberg Trace Formula for PSL(2,ℝ), Volume I. Lecture Notes in Math.548. Berlin-Heidelberg-New York: Springer 1976

    Google Scholar 

  24. Helgason, S.: Differential Geometry and Symmetric Spaces. New York: Academic Press 1962

    Google Scholar 

  25. Helgason, S., Johnson, K.: The bounded spherical functions on symmetric spaces. Advances in Math.3, 586–593 (1969)

    Google Scholar 

  26. Hörmander, L.: The spectral function of an elliptic operator. Acta Math.121, 193–218 (1968)

    Google Scholar 

  27. Kolk, J.: Formule de Poisson et distribution asymptotique du spectre simultane d'opérateurs différentiels. C.R. Acad. Sci. Paris Sér. A-B284, 1045–1048 (1977)

    Google Scholar 

  28. Kolk, J.A.C.: The Selberg Trace Formula and Asymptotic Behaviour of Spectra. Thesis, Rijksuniversiteit Utrecht, (1977)

  29. Lax, P.D., Phillips, R.S.: Scattering Theory for Automorphic Functions. Ann. of Math. Studies No 87, Princeton: Princeton University Press 1976

    Google Scholar 

  30. Mackey, G.W.: The Theory of Unitary Group Representations. Chicago: The University of Chicago Press 1976

    Google Scholar 

  31. Menikoff, A., Sjöstrand, J.: On the eigenvalues of a class of hypoelliptic operators. Math. Annalen,235, 55–85 (1978)

    Google Scholar 

  32. Minakshisundaram, S., Pleijel, A.: Some properties of the eigenfunctions of the Laplace-operator on Riemannian maniforlds. Canadian J. Math.1, 242–256 (1949)

    Google Scholar 

  33. Mostow, G.D.: Intersections of discrete subgroups with Cartan subgroups. J. Indian Math. Soc.34, 203–214 (1970)

    Google Scholar 

  34. Mostow, G.D.: Strong Rigidity of Locally Symmetric Spaces. Ann. of Math. Studies No 78. Princeton: Princeton University Press 1973

    Google Scholar 

  35. Ozols, V.: Critical points of the displacement function of an isometry. J. Differential Geometry3, 411–432 (1969)

    Google Scholar 

  36. Prasad, G., Raghunathan, M.S.: Cartan subgroups and lattices in semi-simple groups. Ann. of Math.96, 296–317 (1972)

    Google Scholar 

  37. Randol, B.: On the analytic continuation of the Minakshisundaram-Pleijel zeta function for compact Riemann surfaces. Trans. Amer. Math. Soc.201, 241–246 (1975)

    Google Scholar 

  38. Randol, B.: The Riemann hypothesis for Selberg's zeta-function and the asymptotic behavior of eigenvalues of the Laplace operator. Trans. Amer. Math. Soc.236, 209–223 (1978)

    Google Scholar 

  39. Schiffmann, G.: Intégrales d'entrelacement et fonctions de Whittaker. Bull. Soc. Math. France99, 3–72 (1971)

    Google Scholar 

  40. Selberg, A.: Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series. J. Indian Math. Soc.20, 47–87 (1956)

    Google Scholar 

  41. Varadarajan, V.S.: Lie Groups, Lie Algebras and Their Representations. Englewood Cliffs: Prentice Hall 1974

    Google Scholar 

  42. Varadarajan, V.S.: Harmonic Analysis on Real Reductive Groups. Lecture Notes in Math.576. Berlin-Heidelberg-New York: Springer 1977

    Google Scholar 

  43. Wallach, N.R.: On the Selberg trace formula in the case of compact quotients. Bull. Amer. Math. Soc.82, 171–195 (1976)

    Google Scholar 

  44. Wallach, N.R.: An asymptotic formula of Gelfand and Gangolli for the spectrum of Γ/G. J. Differential Geometry11, 91–101 (1976)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut, Rijksuniversiteit, Budapestlaan 6, Utrecht, Netherlands

    J. J. Duistermaat

  2. Dept of Mathematics, University of California, 90024, Los Angeles, CA, USA

    J. A. C. Kolk & V. S. Varadarajan

Authors
  1. J. J. Duistermaat
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. J. A. C. Kolk
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. V. S. Varadarajan
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Work partially done at the Mathematisch Instituut, Rijksuniversiteit Utrecht

Work partially supported by NSF Grant MCS 76-05962. In addition, this author would like to acknowledge his gratefulness to the authorities of the Mathematisch Instituut, Rijksuniversiteit Utrecht and the Mathematisches Institut, Universität Bonn, for their hospitality during the Fall and Winter of 1975, when a part of this work was carried out

Rights and permissions

Reprints and permissions

About this article

Cite this article

Duistermaat, J.J., Kolk, J.A.C. & Varadarajan, V.S. Spectra of compact locally symmetric manifolds of negative curvature. Invent Math 52, 27–93 (1979). https://doi.org/10.1007/BF01389856

Download citation

  • Received:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF01389856

Keywords

Search

Navigation